# Does exchanging integer variables by binary variables strengthen a MIP?

Suppose that I have an MIP with a whole lot of continuous variables and some integer variables. In my case, this takes a very long to solve (in fact I wasn't able at all to solve it to optimality). So I was wondering if it would be beneficial for solving this problem to exchange my integer variables by multiple binary variables.

The integer variables do have upper bounds in the range of $$10^2$$, so I would replace them with ~100 binary variables. Does this help my LP relaxation to get stronger? Actually, I would suggest to slightly adjust objective function coefficients, so I don't end up with ~100 variables of the exact same nature (I guess this might help).

I would appreciate any experience or expertise on this topic, Cheers.

• This question is very relevant. Jul 13 at 11:13
• A mere replacement of integer variables with their binary equivalents will likely have no effect (the model could be solved a little faster or slower due to the change in the branch and bound search tree). If you could utilize binary variables (e.g., use them to derive valid inequalities, etc, as mentioned in the accepted answer to this question) then it might help to use binary variables. Jul 13 at 11:16
• if you post the whole formulation it is easier to judge Jul 13 at 19:34

$$\sum_{i=0}^{\left \lceil{\log_2(10)}\right \rceil } 2^i\tilde{x}_i \text{ subject to } \sum_{i=0}^{\left \lceil{\log_2(10)}\right \rceil } 2^i\tilde{x}_i \leq 10$$